DIMENSIONAL REGULARIZATION AND THE RENORMALIZATION GROUP G. 't HOOFT * CERN, Geneva

Received 8 May 1973 Abstract: The behaviour of a renormalized field theory under scale transformations x --* kx; p -'* p / h can be found in a simple way when the theory is regularized by the continuous dimension method. The techniques proposed here have several applications in dimensionally regularized theories: short distance behaviour is expressed in terms 9f the single poles at n = 4, and all coefficients in front of the higher poles 1/(n - 4)g are expressed in terms of those of the single poles l [ ( n - 4), without calculating any diagram. In some cases the Taylor series in 1/(n - 4) can be summed, leading to some exact results for the infinities of the full theory.

1. Introduction The continuous dimension method has recently been advocated by many authors as a useful device for obtaining a finite perturbation expansion of a renormalized field theory [ 1, 2]. As its main advantages one usually mentions the facts that no additional regulator diagrams are needed that would make the algebra more complicated, and that complicated symmetries like local gauge invariance are left intact [ 1] In this article we would like to point out still another useful aspect: it becomes rather transparent how the theory behaves under spacetime scale transformations. This behaviour is described by a simple differential equation, closely related to the Callan-Symanzik equation. Our technique cuts both ways: we also find an important set of equations between the residues of different poles at n ~ 4.

2. The role of the unit of mass in the subtraction procedure Let ~ (~0i, X) be a Lagrangian for a renormalizable field theory with fields ~oi, in which k is an expansion parameter, for instance a coupling constant. The continuous dimension method consists in considering the theory in a "space* On leave from the University of Utrecht, Utrecht.

456

G. 't Hooft, Dimensional regularization

time" with dimension n :/= 4. For all integrals that can occur in the perturbation expansion, one can define in a unique way its finite part as an analytic function of n. If we now consider the limit n -~ 4 we encounter poles of the type (n - 4 ) - g , which can only be removed if we let the coefficients in ~2 also vary as a function o f n . Thus the Lagrangian 52 (~oi, X) is replaced by 52= 52(~.0i,~) + A~2 (~oi, X , n ) ,

A( i) + x2[B( i) A52= X n _ 4

Ln-4

] +(n_4) 2-1+''"

(2.1)

The Lagrangian (2.1) is constructed such that all Green functions remain finite as n -* 4, order by order in the perturbation expansion in k. Now, if the Lagrangian 52+ A52 can be obtained from ~, itself just by rcnormalizing the parameters and rescaling the fields (and, if necessary, performing gauge transformations), then the theory is called renormalizable. We shall only consider that case. The S-matrix will depend only on those parameters which are invariant under field transformations or gauge transformations. Let us first consider these parameters and let us furthermore assume, for the time being, that there are only two of them: a mass parameter m, and a coupling constant X, occuring in the form of terms like - lm2~02

and

14 X~04

(2.2)

in the Lagrangian. The more general case o f an arbitrary number of parameters will be discussed in sect. 5. In this section we shall not use the particular form of the terms (2.2) except that they define the dimension of these parameters: if we assign to a derivative 0 r dimension 1, the Lagrangian has dimension n. A boson field ~ohas therefore dimension (n - 2)/2. A mass m has dimension 1 and X in (2.2) has dimension n -- 4 - ~ =

4-- n .

Now in the theory at non-rational n, all divergent integrals in the pertubation expansion can be redefined in terms of convergent integrals in a unique way [1 ]. This procedure conserves all possible symmetry aspects of the original Lagrangian, including its scale transformation properties. So, at non-rational n the parameters m and X in (2.2) have a well defined and unique meaning, and their dimension is not influenced by the interactions*. We therefore call these parameters the bare * Note that our subtraction procedure at n ~ 4 differs in a crucial way from that of Wilson, tel [3], who uses a cut-off A, whereas we first redef'me divergent integrals. This is why Wilson gets anomalous dimensions also at n ~ 4 where we do not.

457

G. 't Hoot, Dimensional regularization

parameters which we shall denote with a suffLx B: XB and m B. However, as stated above, we have to let these bare parameters go to infinity while n --, 4. This we do by expressing them in terms o f two Finite (but rather arbitrary) parameters ~'R and mR, and n (R standing for "renormalized"). We shall choose ~'R and m R to be dimensionless and independent o f n, although we may substitute, if we wish, a function that depends in a given way, but always smoothly, on n. To express XB and m B in terms of ?~R, m R and n we need a unit of mass, #, that is kept fixed. So, in practice, we shall have an expansion in terms of hR of the following type:

f

~B =/'t4-n LXR +

al 2(mR)~,2R n- 4

+

al3(mR)X3R n- 4

+ . . . -t

a23(mR)X 3 (n_4)2

-t a 2 4 ( m R ) h 4 ÷ . . . +a34(mR)X4 + . . . + . . . J (n -- 4) 2

m B = # [mR + I_

(n -- 4) 3

'

b22(mR)X2 b l l ( m R ) A R bl2(rnR)?~ 2 + n-4 n--4 +''" + (n--4) 2

+'''+'''1

" (2.3a) lit

The coefficients avi, bvi can be calculated [1, 2, 4] from the Feynman diagrams of order i in ~. If we want not only the S-matrix, but also Green functions to remain finite at n ~ 4, we also must renormalize fields and gauge parameters. Our methods will apply to these renormalizations also (see sect. 5). It will be more convenient not to expand in terms o f ?~R, and to write eqs. (2.3a)

as, ~,B =/a 4 - n [~R + ~ v=l m B =/.t [ m R +

av(mR'hR)q , (n-4) v

-J

~ bv(mR'hR)]. v=l ( n - - 4 ) u

(2.3b)

Now, o f course, eq. (2.3) is not the only expression for ?'B and m B which will lead to a f'mite S-matrix as n --> 4, because one can always substitute k R -~ X R + e l ( n - 4) + e2(n - 4) 2 . . . . m R - , m R + f l ( n - 4) + f 2 ( n - 4 ) 2 . . .

,

(2.4)

so that we get a different series of the kind * Actually, the coefficients a are independent of m R and b are proportional to m R (see sect. 4), but for sake of clarity we do not want to use that information here.

G. "tHooft, Dimensionalregularization

458

;kB =/'t4-n

'

Ik~l =

mB=

'

'

cg(mR, XR) (n

--

4)k + ~'R +

=

Ik~_l d'k(mR,X'R)(n--4)k+m'R+ ~ -

v---(mR')~R (n -- 4) v -J ' v ~1 a' ' ')]

v=l

b'v(mR'XR) 1 ~n~4)

v

(2.5)

-I

This corresponds to the usual ambiguity in any renormalization formalism, but it is easy to convince oneself that the subsidiary condition that all coefficients ci and di in (2.5) be zero makes the expansion (2.3) unique, (once we have chosen a value for #), because two different ones cannot possibly both yield an S-matrix that is finite, order by order in the perturbation expansion. We shall from now on assume that all necessary transformations of the type (2.4) have been performed such that this subsidiary condition holds, so that we have series like (2.3) and not (2.5)*. On the one hand this requirement also leads to a definition of kR and m R in a given theory (in general they do not correspond to the more usual definition like on-massshell coupling constant or physical mass), but on the other hand it will be clear that this definition will depend on our "unit of mass"/.t.

3. The scaling properties o f k R and m R Suppose we calculate a diagram with loops that is ultra-violet divergent but does not become infra-red divergent as m -* 0. After application of our regulation technique and insertion of the series (2.3) the result will contain powers of log~,

(3.1)

where k is a typical external m o m e n t u m and/2 is our "unit of mass". It will be clear that the perturbation expansion will only be applicable if (3.1) is resonably small, that is, we must choose/2 to be of the order of k. If we want to study limits like k 2 -* oo or k 2 -* 0, it is of importance to vary/a also. So it is of importance to compare the same theory at different choices of/a. Our result will be the discovery of a well known fact [5] in a new way: the anomalous scaling behaviour of renormalizable theories. Consider a new unit of mass, #', somewhat larger than/z: /a' =/a(1 + e),

l el < 1,

(3.2)

and express XB and m B in terms o f the new unit: * For practical calculations this condition for the counter terms is probably not the most convement one, due to the appearance of numbers lake the Euler constant "yin the higher coefficients.

G. 't Hoofl, Dimensional regularization ?~B=(la')4-n[l+e(n--4)]

=(#')4-n

[XR + ~ ' av v=l (n - 4) v

459

]

I e ( n _ 4 ) X R +?~R + e a l + ~ av+eau+l] v= 1 "(n -- 4) v _l'

(3.3)

.]

(3.4)

and similarly mB =/~,[1 _ e] [mR + u=~l

bu

(n -- ;~)~-, "

Note that the series (3.3) is not of the desired type because of the occurence of a term proportional to n - 4. We have a series of the type (2.5) instead of (2.3). This can be cured by substituting for kR a slightly n-dependent quantity, that is, by performing a transformation of the type (2.4): ~'R = ~ R -- e ( n -- 4)~" R .

(3.5)

The series (3.3) and (3.4) then become (note that also a and b depend on ~,):

~'B=(/a')4-n ['~R

+ca1 -- eX~RaI,L

+ ~ av(mR"~R)+eau+l--e~"Rav+l'h ]. , v=l

(3.6)

(n - 4) v

and

mB=la,[m

R--emR

_

~ bv(mR,'~R)-ebv-ebv+l, xl , e)k'Rbl, x +__ v=l

(3.7)

(n --4) v

where a~, x stands for aav(m, ~'R)/B ~'R, and similarly by, x, av, m, by, m- Now, eqs. (3.6) and 3.7) have the desired form of eq. (2.3). Indeed, the values of XR and m R have changed in a non.trivial way. We have to define

X~ =~'R +e(al -- ~'"Ral,h) , P

mR = mR -- e(mR + "~Rbl, x)-

(3.8)

In sect. 2 we admitted that ~'R, mR may depend explicitly on n, but all physically relevant quantities are smooth functions of 3,R, m R and n, and, therefore, only the values ofX R and m R at n = 4 are relevant. At n = 4 we have ?~R = 3'R, so we obtained the desired transformation properties of X and m:

460

G. "t Hooft, Dimensional regularzzation

If we change our unit o f mass # into g' = #(1 + e), then we have to change our renormalized parameters XR and m R respectively into XR=X R + e

I1

_X R a

al(mR,XR),

(3.9)

m'R=mR--e[mR+XR~-~Rbl(mR,XR)],

in order to obtain the same renormalized S-matrix following the same prescription of sect. 2. Note that only the residues of the single poles of XB and m B contribute to these scaling properties of XR and m R. Eqs. (3.9) are in fact differential equations for XR and m R as a function of/a. Examples of such equations will be discussed in sect. 6.

4. Identities for the coefficients a v and b v We have not yet written down the complete series that replaces (2.3) after a scale transformation/a ~/a'. The substitution that has to be made in (3.3) is the product of (3.5) and (3.8): F

t

XR = ~R -- e(n -- 4)~, R + e ( - - a 1 + ~,Ral, x) ; t

I

t

(4.1)

m R =m R + e(m g +XRbl,x).

We get from (3.3): ~,B = (/a')4-n [XR + ~ 1 ' ' , v=l (n -- 4) ~ ( a v ( m R ' x R ) + eav+l -- eXRav+l' x

,

,

+ ear, x ( - a l + h R a l , x) + ear, m(m'R + XRbl, x))

(4.2)

]

,

and m B =#

[

mR +

v= 1 (n -- 4) v t

(bv(m'R,X'R) -- eb v - e X a b v + l , h

(4.3) t

t

+ ebu, x ( - a l + XRal, x) + e b u , m ( m R + XRbl, x)

"11

q

o

Note that a v, b v, m R and ~kR are all dimensionless. Obviously, the functions a v and b v are independent of/a.

G. 't Hoofl, Dimensional regularization

461

But, as has been argued in sect. 2, the series (2.3) is the only series that leads to a finite S-matrix at n -~ 4. Hence the series (4.2) and (4.3) must be the same as (2.3), so the "correction terms" proportional to e must all cancel. Those terms proportional to e that are of zeroth order in 1/(n - 4), have been made to vanish by construction, but for the higher order ones this observation leads to important identities between the coefficients av, by:

av, h(--al + XRal, h) + av, m (mR + ?'Rbl, ~,) = XRav+ I, h -- av+ 1 ;

(4.4a)

by, h ( - a l + XRal, x ) + bv, m (mR + ?'Rbl, h ) -- bv = XRbv+ I, h "

(4.4b)

This is the set o f equations we alluded to in the beginning. For instance, it follows that in (2.3a) aal2 =0 ; ~rn R

(4.5a)

2a23 = 2a22 + m R

aal3. , 8m R

(4.5b)

abll mR ~--~R = b l l ;

(4.5c)

b121 abl2 2b22=~12bll + mR + mR - ~ R -- b12"

(4.5d)

and so on. We see that a12 is independent o f m R and b l l is proportional to m R. Now, one can show [1 ] that all counter terms in the dimensional regularization procedure are not only polynomials in terms of external momenta but also in terms of masses m R. In particular, there is no singularity at m R ~ 0. Given this fact, it is not difficult to derive from eqs. (4.4) that all coefficients aii are independent o f m R and all coefficients bii are linear in m R, so the last term in (4.5b) and the last two terms in (4.5d) may be dropped; likewise one can then substitute av, m = 0; by, m =bv/m R in (4.4). A qualitative interpretation of(4.4) can then perhaps be made: the subdivergences of a diagram are expressed in terms of over-all divergences and subdivergences of subgraphs. We have not checked the possibility of this interpretation in detail. The validity of (4.4) has been checked in some exampies. Eqs. (4.4) may be reformulated in terms of a differential equation for the functions a(XR, m R, n) = 2.a ~=l

av(XR , m R )

,

etc.

(n--4) ~

This equation, for the X~o4 theory, will be discussed in sect. 7.

G. "t Hoofl, Dimensional regularization

462

5. Generalization towards an arbitrary number of parameters

The preceding section was meant to illustrate our arguments in detail for the case of only two different kinds of parameters. Let us now take a renormalizable theory with an arbitrary but fLxed number of parameters ;~k, which in a bare Lagrangian have dimension D k, D k = p(k)(4 -- n) + O(k ) .

(5.1)

Again we write

.--,

(5.=)

In gauge field theories also renormalization of gauge parameters a k may be needed: o

k +

.

v=l

(5.3)

" ( n - - 4 ) v .z

Renormalization of the fields can also be considered: k

Dk

(5.4)

+

Both a R and ~R do not enter into (5.2) so the physical parameters •R will satisfy equations among themselves. In order to fred the scaling behaviour, or the pole equations, for a R and ~1~ one simply absorbs (5.3) and (5.4) into (5.2). From now on we assume that that may have been done. Consider U' = U(1 + e) ;

(5.5)

the equivalent of eq. (3.3) is Xk = (/a')Dk [ eP(k)(n -- 4)k k + k k + eP(k)a~l -- eO(k)Xk (k)

+

v= 1

(k)

(n - 4) ~

(5.6)

.l ]

Let us write ~k

= ~.,k _eP(k)(n_a)x,k

+sxk,

(5.7)

and substitute that into eq. (5.6). The term 8 Xk is required to be such that the first term of the series in 11(n - 4) F then obtained, will be just XR; hence

463

G. 't. Hooft, Dimensional regularization

,l

8XkR = ea(k)X ,kR _eP(k)a~l + ~ e a k ,

lP(l)X R

(5.8)

.

1

where a k, l stands for (8/~ X/R)ak(X R ). Now

hk = (P")Dk [ ~''k+ z,~l ~,nl_'4)v{ak(~'')-e°(k)ak+e'(kak+l) °

-~e,,ak'~(o ~,(1>u+l,,?~ IR +e

l

ak,, (l)XIR--P(1)a{

tm

(5.9)

m

Putting n = 4 in eq. (5.7) we find the scaling properties of the parameters ?~R:

~' = ~(I + e), (5.10) I

and the generalization of eqs. (4.4) becomes

~ a k l(o(1)~I --p(1)a~+ D a~,mP(m)X~)-- O(k)ak I m =

a kv+ l,lP(1)hlR 1

k _ p(k)au+l •

(5.11)

In most theories the parameters with o(k) = 0 will only get counter terms that are independent of the other parameters. This observation leads to a simplification similar to the one in the end of sect. 4.

6. Scaling behaviour of some theories Let us define a scaling parameters s by

= ~o e' ,

~

= X~(s).

Then we can rewrite eqs. (5.10) in terms of a differential equation:

(6.1)

464

G. 't Hooft, Dimensional regularization

ds -

O(k);kk + P(k)akCAR ) -

~ a k1, l (XR) P(/) XR l • l

(6.2)

This equation is nearly, but not quite, identical to the Callan-Symanzik equation [5]. Here the coefficients are expressed in terms of the dimensions and the poles of the parameters. Let us again consider the case that there is only one parameter X with o = 0, as in X~04 theory. Then a 1 only depends on this parameter (see sect. 4) and we have a (nonqinear) first order differential equation with one variable. Using the notation of (2.3a) and dropping the suffix R, we get d~ dal ~ - = al(X ) - 3,--~- = -a12X2 + O(X3).

(6.3)

It will be clear that the behaviour of the theory under scale transformations depends crucially on the relative sign of ?, and a12. Two examples of one-parameter theories are well known: (i) the scalar theory with coupling -X~04. The sign of X is usually taken to be positive, in order for the Hamiltonian to be positive definite*; (ii) quantum electrodynamics. Here X is to be replaced by e 2, which is always positive. Both of these theories have a12 < 0 which implies that XR(S) diverges as s~ +~. This implies that the small distance behaviour of these theories is not described by the usual perturbation expansion. The long distance behaviour on the other hand, can be found quite accurately because for s -~ -*% X -* 0 and the first terms of the perturbation expansion converge rapidly there. In this region we can give the solution of eq. (6.3): 1

~R(S) - C + al2s

(6.4)

eq. (6.4) may be used to fred infra-red behaviour for massless theories, but in general has no meaning ff massive particles occur, as a consequence of "ultra-violet" divergencies at m2 [l~2 -* oo. In a pure Yang-Mills theory, also, ?~is to be replaced by g2, where g is the usual coupling constant. So X > 0. However, in that theory a12 > 0 [4, 7] and the solution (6.4) is valid in the ultra-violet region s ~ +oo. The same behaviour is found in X~04 theory if X is taken the unusual sign [6]. A Yang.Mills theory with fermions can be written down [4, 7] with the property a12 = 0. In this theory the two-loop diagrams are decisive for its scale behaviour. Suppose a13 ~: 0. A small change in the theory can make a12 very small but not zero. In that case the function a 1 - X(~al/~;~ ) in (6.3) must have a zero for a small * The correctness of this axgument is dubious. See ref. [6] and the last paragraph of this section.

465

G. "t Hoofl, Dimensional regularization

value ?to o f h. We then have a conformally invariant theory with a small coupling constant, so that both the perturbation expansion and the conformal invariance can be studied. The situation becomes quite complicated if we have two or more parameters h i with off) = 0 In general, eq. (6.3) will be replaced by

d, -

a,kh

hi+ O(X3)

(6.5)

where a!, IK are determined by the poles of the one-loop graphs In many cases the solutions diverge at both ends o f the s-scale [7]. Neither the ultra-violet, nor the infra-red behaviour can then be calculated. In a certain example we also found that a parameter XR can easily change sign, which is one more reason to doubt the argument that the corresponding classical Hamiltonian must be positive definite.

7. Renormalization and the perturbation expansion In this section we show an application of eq. (5.11). Let us again consider eqs. (2.3) and now write l__k__ -

hR + = av(kR) (n -- 4) v - kBCAR' n ) ,

(7.1)

where we confine to those theories for which a u are independent of m R (like in ~k tO 4 t h e o r y , s e e s e c t . 4). Putting also (7.2)

a0 = hR , eq. (4.4a) becomes aa v aXR ( - - a l + hRax, h) = h R 0%+1 ah R for v = 0, 1,2 . . . . .

(7.3)

av+l ,

or

~XB [ ah B a h R A ( h R ) + (n -- 4) thR ¢3--~R-RR-- hB) = 0 .

(7.4)

where A (hR) stands for a t - hRal, h. The general solution o f eq. (7.4) is XB(hR, n) = exp

+ Cx(n)

dh Co

X + (n - 4 ) - 1 A CA)

}.

(7.5)

G. 't Hooft, Dimensional regularization

466

The integration path in ~,-space may be arbitrary. We must take in mind that A(k) is for small k of order ~2, so we write: A(~) = X2AQ,),

(7.6)

and XB =XR exp

{/S --

dX

0

1 (n - 4 ) 5 - X ( x )

+C2(n )

}

.

(7.7)

+x

According to (7.2) we require that XB -~ XR + O(X2) i f k g ~ 0. It foUows that in eq. (7.7) C2(n) = 0.

(7.8)

Strictly speaking eq. (7.7) with (7.8) should be interpreted as follows: expanding it with respect to XR one obtains a series of which each term has a certain singularity at n = 4. Substituting that series in the expansion of the S-matrix in terms of )'B (which also has singularities at n = 4) one obtains as a result a series for the S-matrix, in which each term is finite at n = 4. But it is appealing to assume that if a perturbation expansion contains only finite and well-defined terms, then the full theory will also be finite. In that case one merely needs to substitute eq. (7.7) itself into the "full theory" at n :/= 4, to obtain a finite theory at n = 4. Of course, we have no means to check such an assumption, but it is natural and let us see its consequences. At n -~ 4, eq. (7.7) behaves like (ifA (0) 4: 0):

1 ~'(0) ~'B = - ~ ) ( n -- 4) + .~A...(0))3(n__- 4) 2 log (n - 4) + RI(n,XR)(n - 4 ) 2 ,

(7.9)

where Rx(n, XR) is a function that behaves smoothly at n --*4, and Rl(4, XR) depends on k R:

1

R1(4 , kR) "" -- (/T(0))2 Xg + 0 (log XR),

(7.10)

for small ~RSo here is another remarkable result for renormalization theory: If the bare coupling constant kB ih ~4 theory (or g2 in pure Yang-Mills theory) is given an n dependence as in eq. (7.9), then the theory is f'mite at n -~ 4. Note that we may give 3,R a smooth n dependence, so R l ( n ) is an arbitrary function of n, finite at n = 4. The numbers 5 ( 0 ) = -- a12

and

/T'(0) = -- 2 a13,

(7.11)

(see eq. (2.3a)), correspond to the one-loop and two-loop poles respectively, and can be calculated exactly. For instance

G. 't Hoofl, Dimensional regularization

~(0)-

3

16 ~r2

467

(7.12)

in ~4 t h e o ~ (with the usual sign for L) and .4(0) -

11

,

(7.13)

12 rt2 for the coupling constant g2 in pure SU(2) Yang-Mills theory. Although 2/B ~ 0 we have not a free field theory (the interaction strength is roughly equal to ~R in eq. (7.10)). Note also that our result, formula (7.9), only holds for the "full" theory, not the perturbation series, as can be deduced from the singular behaviour of R 1(4, ~tR) at ~lR ~ 0 (see eq. (7.10)). In the same way the mass-renormalization can be calculated. Taking bv(~t) proportional to mR, or m B = p(XR, n)m R ,

we find from (4.4b):

p('AR, n) = exp

~ 0

R

~-(~,) dh hA(k) + n - 4

'

(7.14)

where ;tbl, x = mR XB (~/) • Here also one could consider first taking n ~ 4. We f'md that finiteness of the theory then requires the mass renormalization (ifA (0) ~e 0): m B = (n - 4) -/~(0)/X(0) R 2 ( n ) ,

(7.15)

with R2(n ) finite at n ~ 4, and proportional to m R. Note that B (0) = b l l / m R (see (2.3a)), corresponds to the one-loop mass renormalization and can be calculated exactly also. In the ~ktp4 theory B"(0) -

1 321r 2 '

(7.16)

and eq. (7.15) becomes 1

m B = (n - 4) ~ R2(n).

(7.17)

It is remarkable that only one-loop inf'mities contribute to this mass renormalization, while only the one and two-loop infinities determine the coupling constant renor-

468

G. 't Hooft, Dimensional regularization

malization. It is left as an exercise for the reader to see what happens in a theory with A (0) -- 0.

8. Conclusions Making use of the observation that at n ~ 4 all integrals in perturbation field theory can be made finite, and that these finite expressions have the "naive" scaling properties, we found that the scaling behaviour at n = 4 of dimensionally regularized field theories can be formulated in terms of simple equations (3.9), (5.10), (6.2). These equations are closely related to the Callan-Symanzik equations [5], but they have the advantage that the coefficients are completely determined by the residues of the poles at n = 4, which one adds to the parameters in the Lagrangian, in order to make the theory finite at n ~ 4. In particular the single-loop contribution to these poles (which are the most important ones) can be obtained rapidly for complicated theories by means of a "pole-algebra", to be derived in a future publication [4]. Further we derived equations between residues of lower and higher poles at n = 4, which are so stringent that in certain cases the complete singular behaviour at n -* 4 of the bare parameters in the Lagrangian can be determined exactly (eqs. (7.9), (7.15)). The results in sect. 7 should, however, be interpreted with care, since they hold only for the summed theory (if such a thing exists), not for the individual terms in the perturbation expansion.

References

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